Theorem raleqbidvv 3330

Description: Version of raleqbidv 3338 with additional disjoint variable conditions, not requiring ax-8 2146 nor df-clel 2839. (Contributed by BJ, 22-Sep-2024.)

Hypotheses

Ref	Expression
raleqbidvv.1	⊢ (𝜑 → 𝐴 = 𝐵)
raleqbidvv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
raleqbidvv	⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem raleqbidvv

Step	Hyp	Ref	Expression
1		raleqbidvv.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		raleqbidvv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
3	2	adantr 484	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒))
4	1, 3	raleqbidva 3328	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐵 𝜒))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562   ∈ wcel 2144  ∀wral 3078

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1802  df-cleq 2756  df-ral 3079  df-rex 3089

This theorem is referenced by:  raleqbi1dv  3332  dfttc4lem2  36894  gpg5nbgrvtx03star  48707  postcposALT  50194